NLS blowup: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
No edit summary
 
No edit summary
Line 3: Line 3:
<br /> In the <math>L^2\,</math>-supercritical focussing [[NLS]] one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality
<br /> In the <math>L^2\,</math>-supercritical focussing [[NLS]] one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality


<center><math>\partial^2_t \int x^2 |u|^2 dx ~ H(u)</math>;</center>
<center><math>\partial^2_t \int x^2 |u|^2 dx \leq H(u)</math>;</center>


see e.g. [[OgTs1991]]. By scaling this implies that we have instantaneous blowup in <math>H^s\,</math> for <math>s < s_c\,</math> in the focusing case. In the defocusing case blowup <br /> is not known, but the <math>H^s\,</math> norm can still get arbitrarily large arbitrarily quickly for <math>s < s_c\,</math> [[CtCoTa-p2]]
see e.g. [[OgTs1991]]. By scaling this implies that we have instantaneous blowup in <math>H^s\,</math> for <math>s < s_c\,</math> in the focusing case. In the defocusing case blowup <br /> is not known, but the <math>H^s\,</math> norm can still get arbitrarily large arbitrarily quickly for <math>s < s_c\,</math> [[CtCoTa-p2]]

Revision as of 21:00, 6 August 2006



In the -supercritical focussing NLS one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality

;

see e.g. OgTs1991. By scaling this implies that we have instantaneous blowup in for in the focusing case. In the defocusing case blowup
is not known, but the norm can still get arbitrarily large arbitrarily quickly for CtCoTa-p2